Event quests are solo or Co-Op quests available for limited time frames. Solo Event quests cycle weekly that drop increased amounts of each Material.
Solo Event quests are available by selecting "Quest" on the Home screen, then selecting "Event" at the bottom right.
Co-Op Event quests are available by selecting "Co-op Battle" from the Home screen, then selecting the currently running Event.

Co-Op quests allow you to select 2 characters to team with 2 characters of another player.
Co-Op Quests have a higher chance of dropping certain B-Rank (or higher) equipment than the solo quests.
The currently available Co-Op Quest lines are: